Frequency-Domain Numerical Modelling of Visco-Acoustic Waves Based on Finite-Difference and Finite-Element Discontinuous Galerkin Methods

Brossier, Romain; Etienne, V.; Operto, S.; Virieux, Jean

doi:10.5772/9714

Cited by 27 publications

(32 citation statements)

References 69 publications

(68 reference statements)

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“…A first simulation is performed for an attenuation Q = 10000 to compare the hybrid solver modelling solution with the acoustic solution of the time-domain approach. The hybrid solver modelling simulation is done with the weights of the mixed grid stencil corresponding to G m = 4, 6, 8, 10 (Brossier et al 2010). A qualitative comparison between the monochromatic wavefields computed with the hybrid solver modelling and the time-domain approaches shows a reasonable agreement given the coarse parameterization used for both approaches (4 grid points per minimum wavelength) and the different numerical stencils used for the two modellings ( Fig.…”

Section: Validation Against Finite-difference Time-domain Solutionsmentioning

confidence: 90%

“…For 3D problems, the time and memory complexities of the direct solvers dramatically increase (O(N 6 ) and O(N 4 ), respectively, where N denotes the size of a N 3 cubic grid) and limit the size of the applications that can be performed with such approaches (by complexity we mean the increase of the computational cost of the modelling with the size of the problem) (Operto et al 2007). However, Brossier et al (2010) recently showed that visco-acoustic modelling can be efficiently performed with the MUMPS direct solver (MUMPS-team 2009) in the SEG/EAGE overthrust and salt models for frequencies of the order of 7 Hz on a limited number of message passing interface processes with a significant amount of shared memory per process (15 Gbytes). A new approach, which exploits the low-rank property of the Helmholtz equation for compression, has also been recently proposed to mitigate the memory and time complexities of the LU decomposition performed with the multifrontal method (Wang, Xia and de Hoop 2010).…”

Section: Introductionmentioning

confidence: 99%

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Three‐dimensional parallel frequency‐domain visco‐acoustic wave modelling based on a hybrid direct/iterative solver

Sourbier

Haidar

Giraud

et al. 2011

Geophysical Prospecting

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Special Issue: Modelling Methods for Geophysical Imaging: Trends and PerspectivesInternational audienceWe present a parallel domain decomposition method based on a hybrid direct-iterative solver for 3D modelling of visco-acoustic waves in the frequency domain. The modelling method was developed as a modelling engine for frequency-domain full waveform inversion. Frequency-domain seismic modelling reduces to the solution of a large and sparse system of linear equations, resulting from the discretization of the heterogeneous Helmholtz equation. Our approach to high-performance, scalable solution of large sparse linear systems in parallel scientific computing is to combine direct and iterative methods. Such a hybrid approach exploits the advantages of both direct and iterative methods. The iterative component allows us to use a small amount of memory and provides a natural way for parallelization. The direct part provides its favorable numerical properties. The domain decomposition is based on the algebraic Schur complement method, which allows for the iterative solution of a reduced system, the solution of which is the pressure wavefield at the interfaces between the subdomains. Once the interface unknowns have been computed, the wavefield at the interior of each subdomain is efficiently computed by local substitutions. The reduced Schur complement system is solved with the global minimum residual method (GMRES) and is preconditioned by an algebraic additive Schwarz preconditioner. A direct solver is used to factorize the local impedance matrices defined on each subdomain. Simulations are performed in the overthrust and the salt models for frequencies up to 12.5~Hz. The numerical experiments show that the number of iterations increases linearly with the number of subdomains for a given computational domain but that the elapsed time of the iterative resolution remains almost constant. The number of iterations also increases linearly with the frequencies, when the grid interval is adapted to the frequencies and the size of the subdomains is kept constant over frequency. Although the hybrid approach allows one to tackle larger problems than the direct-solver approach, further improvements are needed to mitigate the computational burden of the iterative component of the hybrid solver within the framework of multisource modelling. On the numerical side, the use of block iterative solvers and of incremental two-level deflating preconditioners, and on the parallel implementation side the use of two levels of parallelism in the domain decomposition method should allow us to mitigate this computational burden

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Section: Validation Against Finite-difference Time-domain Solutionsmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

Three‐dimensional parallel frequency‐domain visco‐acoustic wave modelling based on a hybrid direct/iterative solver

Sourbier

Haidar

Giraud

et al. 2011

Geophysical Prospecting

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“…The vectors w ¼ ðv x ; v z ; p; qÞ and s A ¼ ðbf xδ ðx − x s Þ; bf zδ ðx − x s Þ; ιωp 0δ ðx − x s Þ; 0Þ are the monochromatic velocity-stress wavefields and source, respectively. We discretize the velocity-stress wave equation A-2 with a nodal formulation of the discontinuous Galerkin method, based on Lagrange polynomials of order 0, 1, or 2 (referred to as P0, P1, and P2, respectively) to perform seismic modeling in VTI acoustic media (Brossier et al, 2010a;Brossier, 2011). A second-order wave equation for particle velocities, useful for FWI implementation, can be inferred from the velocity-stress wave equation A-2 by eliminating the stress wavefields p and q in the first and second rows as follows:…”

Section: Appendix a Acoustic Vti Modelingmentioning

confidence: 99%

Which parameterization is suitable for acoustic vertical transverse isotropic full waveform inversion? Part 1: Sensitivity and trade-off analysis

et al. 2013

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In most geologic environments, accounting for anisotropy is necessary to perform acoustic full waveform inversion (FWI) of wide-azimuth and wide-aperture seismic data because of the potential dependence of wave speeds on the direction of the wave propagation. In the framework of multiparameter FWI, the subsurface parameterization controls the influence of the different parameter classes on the modeled seismic data as a function of the scattering angle and hence the resolution with which the parameters can be reconstructed and the potential trade-off between different parameters. We have evaluated a numerical procedure based on computation of the scattering patterns of the different parameters to assess the sensitivity of the seismic data to different parameterizations of vertical transverse isotropic media in the acoustic approximation. Among the different categories we have tested, a monoparametric FWI was found for imaging one wave speed with a broad wavenumber content, keeping the Thomsen parameters fixed, which have a small influence on the data relative to the wave speed. This raises the question of the initial information required in the background models of the Thomsen parameters to perform reliable monoparameter FWI. Alternatively, simultaneously inverting the horizontal and vertical wave speeds introduces limited trade-off effects because these wave speeds have significant influence on the data for distinct ranges of scattering angles, while the influence of the Thomsen parameter δ remains weak. With such parameterization, the short-to-intermediate wavelengths of the vertical velocity are updated from the short-tointermediate scattering angles, while the long-to-intermediate wavelengths of the horizontal velocity are updated from the wide-to-intermediate scattering angles. We concluded that the choice of the subsurface parameterization can be driven by the acquisition geometry, which controls the scattering-angle coverage and hence the resolving power of FWI, and by the accuracy of the available initial FWI models.

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“…An optimized anti-lumped-mass mixed grid stencil provides the accuracy of a O(Δx 4 ) scheme, although it relies on a linear combination of compact O(Δx 2 ) stencils , Brossier et al, 2010. This compact stencil reduces the fill-in of the matrix during factorization and, hence the memory requirement, while using discretization rule of 4 grid points per minimum wavelength.…”

Section: Toy3dac Codementioning

confidence: 99%

Performances of 3D Frequency-Domain Full-Waveform Inversion Based on Frequency-Domain Direct-Solver and Time-Domain Modeling: Application to 3D OBC Data from the Valhall Field

Brossier

Etienne²,

et al. 2013

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This study addresses the performances of different approaches to tackle 3D frequency-domain Full Waveform Inversion (FWI). FWI is becoming an appealing method to build high resolution velocity models from wide-azimuth seismic data. Three-dimensional frequency-domain solutions of the wave-equation for frequency-domain FWI can be computed with several methods. In the frequency domain, direct or iterative solvers can be used to solve the linear system resulting from the discretization of the wave equation. Alternatively, frequency responses can be computed by time-domain modeling coupled with discrete Fourier transform or phase-sensitive detection method. For a large number of seismic sources, the computational time of all of the methods scale along similar lines to the size of the problem, although the performances of the direct-solver approach is hampered by the limited scalability and the memory demand of the lower-upper decomposition of the forwardproblem operator. We implement the direct-solver and the time-domain modeling approaches in 3D acoustic frequency-domain FWI, which is applied to real 3D wide-azimuth OBC data from the Valhall field, North Sea, at low frequencies (< 5 Hz). We show, for this case study, that even if less flexible for high-performance computing, the direct-solver approach is one order of magnitude more efficient in computing time/resources that the time-domain approach, if the signal-to-noise ratio in the data is sufficiently-high to limit the inversion to few frequencies. We finally discuss the performances, pros and cons of both methods, which depend on the wave physics (acoustic versus visco-acoustic, anisotropy, extension toward elastic or viscoelastic), the acquisition geometry (streamer-like versus fixed-spread acquisitions) and the computing architectures (small versus large amount of memory per computing node). TX 75083-3836, U.S.A., fax +1-972-952-9435

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Frequency-Domain Numerical Modelling of Visco-Acoustic Waves Based on Finite-Difference and Finite-Element Discontinuous Galerkin Methods

Cited by 27 publications

References 69 publications

Three‐dimensional parallel frequency‐domain visco‐acoustic wave modelling based on a hybrid direct/iterative solver

Three‐dimensional parallel frequency‐domain visco‐acoustic wave modelling based on a hybrid direct/iterative solver

Which parameterization is suitable for acoustic vertical transverse isotropic full waveform inversion? Part 1: Sensitivity and trade-off analysis

Performances of 3D Frequency-Domain Full-Waveform Inversion Based on Frequency-Domain Direct-Solver and Time-Domain Modeling: Application to 3D OBC Data from the Valhall Field

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